Implementation

3.3.1 A test problem

A multi-echelon production, transportation and distribution model as shown in fig. 3.2 is employed as a test bed for our method. To keep the study manageable, we restrict our attention to a very basic model. The production facility (PF) produces finished goods to downstream retailers. The retailers face a stationary Poisson demand process with mean inter-arrival time of 1/λ. Machining process is as introduced in section 3.2. Successfully finished goods will leave the machine and go to the next stage for final inspection before shipping to a remote DC. After inspection, any imperfect product has to go back to the processing stage for reworking. Assume that the feedback rate is constant with probability δ. For the sake of simplicity we assume that the second (inspection) stage will never fail.

Products passing inspection will wait at the shipping area, ready for transportation to DC.

Upon arrival at the DC, the product will immediately be transported to the assigned retailer whenever a transporter is available. Again, for the sake of simplicity, we restrict all transporting vehicles between any two sites to one. Assume that all the transportation times are stochastic. Applying the method as described in section 3.2, we formulate this problem as a tandem queue with 5 independent stages. The first stage is the production stage with the unreliable machine being formulated as two “on” and “off” phases. The second stage is the inspection stage. The transportation from PF to DC and the DC itself are formulated as respective M/M/1 queueing systems. Finally, the distribution stage is formulated as a phase type, even though it is easier to accumulate all the retailers as one single stocking site, and

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treat it as an M/M/1 alike, as shown in section 3.2.

Fig. 3.2 A multi-echelon SC with feedback consideration.

Please note that we omitted the other Nj and Ij except those of PF in fig. 3.2.

Specifically, N3, the input queue of the transit from PF to DC; I3, the output buffer of the transit from PF to DC; N4, the input queue of DC; I4, the output buffer of DC, N5, the input queue of the transit from DC to retailer; I5, the output buffer of the transit from PF to DC, which is set to the accumulative retailer inventory level in this design. Further, assume there is an infinite supply at the first stage.

I3 is always zero, assuming the MTO policy is adopted by this service. At DC, it’s reasonable to adopt the MTS policy to lessen the customer order waiting time. Assume that the DC processes its inventory with high efficiency at near zero operation time. This means that each arriving good will be put into stock immediately if there is no backorder recorded.

When there is a backorder, the arriving unit will be shipped to the waiting retailer. N4 is always zero as well. If the customer order arrives, and the stock is out, a situation, which the MTO-type control is sure to encounter, unfilled orders are backlogged and will be satisfied when replenishing goods arrive on a FCFS basis.

3.3.2 Numerical results

Here reports our tests of the approximation of the model illustrated above, and compares its predictions to estimates derived from computer simulation, as illustrated in

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Appendix A.2. Basically we follow the same test approach as reported in Zipkin (1995) with some modifications. The queueing system at PF is just like an open Jackson network.

Thus the λ_i are all identical to λ/(1 − δ), where δ is the feedback rate. All ρι are equal to ρ = λ/[u(1 − δ)]. To test the taxing condition on the performance of the approximation, we fix δ = 0.5. ρ is determined by λ/u. Assume that the mean demand rate for each retailer is 0.25 and that there are four retailers. The combined demand rate is 1. We fix u to be either 2.5 or 4, and thus ρ is 0.8 or 0.5 respectively. Assume mean failure and repair rate to be 0.25 and 2.5 respectively. Assume that the average transportation time is 1/4. We adopt a similar simulation stopping criteria as reported in L & Z and Zipkin (1995). Each run simulates thirty replications of 10 000 time units. Assume there is a holding cost of 0.5 for working-in-process per unit and per unit time, a holding cost of 1 for the end retailer inventory per unit and per unit time, a backorder cost of 10 for unfilled retailer orders per unit and per unit time. Five key performance measures are measured, TC (the total incurred cost of operating the chain, which is equal to 0.5⋅WIP + E[I] + 10⋅E[B], see below), SL (average service level measured in no stock-out probability at the retailer site), WIP (the total intermediate inventory, which is defined as all the working-in-process, inventory level at DC, and all the queues of transit, I1+N2+I2+N3+I4+N5, in this case), E[I] (average retailer inventory, which is omitted for space consideration), E[B] (average retailer backorder). Note that in calculating WIP, I3 and N4 are always zeroes, as described above.

Tables 3.1 and 3.2 summarize the results. Note that the parameter setting of table 3.1 is the same as in Zipkin (1995).

Also notice that the ‘SL’ column is not listed in table 3.1 since they are all zeros. The column labeled Sj is the initial base stock level at the respective stages. The column labeled

‘Sim’ represents the simulation estimates; ‘App’ stands for the approximation, and ‘%Err’

is the absolute percentage error of the approximation compared to the simulation value,

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which is defined as |App – Sim| / Sim × 100%. It is evident that the approximation is quite accurate for table 3.1 with all retailers adopting base stock policies with S5 = 0. Table 3.2 shows the results when all retailers adopt base stock policies with S5 ≠ 0. Also, we adjusted the stock levels for all the other stages according to base stock levels of table 3.1. From table 3.2, we see that when S5 ≠ 0, the accuracy of the matrix approximation method is also satisfactory. From table 3.2 several useful observations can be made. For example, in the case of S5 ≠ 0 with ρ = 0.5, an increasing stock level at different stages, except at the last stage, seems to have the same effect of performance influence. The total cost and WIP levels increase and the service levels increase very limitedly while backorder levels decrease slightly. On the other hand, an increasing stock level at the last stage, i.e., retailer inventory level, does increase the service levels and decreases the backorder level, however it does so at the price of higher total cost, which is due to higher retailer inventory levels.

Table 3.1 Approximation vs. simulation (S5 = 0)

TC WIP E[B]

ρ S1 S2 S3 S4 Sim App %Err Sim App %Err Sim App %Err

0.5 0 0 0 0 30.605 30.177 1.40 1.684 1.668 0.95 2.949 2.93 0.64 0.5 1 1 0 1 13.237 13.019 1.65 2.825 2.8908 2.33 1.182 1.1574 2.08 0.5 3 1 0 1 11.388 10.930 4.02 4.359 4.597 5.46 0.921 0.863 6.30 0.5 1 3 0 1 9.029 9.011 0.20 4.322 4.414 2.13 0.687 0.680 1.02 0.5 1 1 0 3 8.494 8.425 0.81 4.28 4.358 1.82 0.635 0.625 1.57 0.5 1 1 0 5 7.345 7.420 1.02 6.08 6.167 1.43 0.43 0.434 0.93

0.8 0 0 0 0 118.03 125.01 5.91 4.625 4.668 0.93 11.825 12.268 3.75 0.8 1 1 0 1 93.182 97.455 4.59 4.821 4.900 1.64 9.077 9.500 4.66 0.8 3 1 0 1 84.620 83.820 0.95 5.436 5.507 1.31 8.19 8.107 1.01 0.8 1 3 0 1 75.606 81.049 7.20 5.126 5.243 2.28 7.304 7.843 7.38 0.8 1 1 0 3 76.335 80.931 6.02 5.154 5.232 1.51 7.376 7.832 6.18 0.8 1 1 0 5 65.185 66.975 2.75 5.828 5.807 0.36 6.227 6.407 2.89

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Table 3.2 Approximation vs. simulation (S5 ≠ 0)

TC SL WIP E[B]

ρ S1 S2 S3 S4 S5

Sim App %Err Sim App %Err Sim App %Err Sim App %Err

0.5 4 4 0 4 4 9.309 9.512 2.18 0.919 0.994 8.16 10.636 11.079 4.17 0.031 0.029 6.45 0.5 12 4 0 4 4 13.167 13.497 2.51 _{0.922 0.994} 7.81 18.396 19.074 3.69 0.028 0.027 3.57 0.5 4 12 0 4 4 13.248 13.484 1.78 _{0.923 0.994} 7.69 18.596 19.068 2.54 0.026 0.026 0 0.5 4 4 0 12 4 13.252 13.482 1.74 0.923 0.994 7.69 18.607 19.067 2.47 0.026 0.026 0 0.5 4 4 0 4 12 16.969 17.198 1.35 _{0.999 1} 0.10 10.624 11.079 4.28 _{0 0 N/A} 0.5 4 4 0 4 20 24.964 25.194 0.92 _{1 1} 0 10.624 11.079 4.28 _{0 0 N/A}

0.8 4 4 0 4 4 29.772 29.923 0.51 0.649 0.702 8.17 8.125 8.331 2.53 2.347 2.335 0.50 0.8 12 4 0 4 4 23.469 20.958 10.70 0.742 0.836 12.67 13.379 14.532 8.62 1.416 1.075 24.09 0.8 4 12 0 4 4 19.066 19.118 0.27 0.816 0.882 8.09 13.947 14.115 1.21 0.930 0.889 4.44 0.8 4 4 0 12 4 22.175 19.119 13.78 0.813 0.882 8.49 13.744 14.114 2.69 1.026 0.889 13.38 0.8 4 4 0 4 12 23.176 21.999 5.08 0.866 0.885 2.19 8.125 8.331 2.53 0.976 0.888 9.05 0.8 4 4 0 4 20 25.047 23.902 4.57 0.944 0.956 1.27 8.125 8.331 2.53 0.407 0.333 18.08

To conclude, the approximation does seem to work well for all the retailers adopting either MTO or MTS operational strategies with one-for-one replenishment policies. When we incorporate all the stochastic features, including imperfect quality, machine breakdown, random transportation, and random distribution in the system, the degradation of the accuracy is only slight, and is often within the tolerance limits of industrial use. The feedback factor can be treated as capacity loss as concluded in Zipkin (1995).